Abstract
The transition from static to sliding friction is mediated by rapid interfacial ruptures^{1,2,3,4,5} propagating through the solid contacts forming a frictional interface^{6}. While propagating, these ruptures correspond to true shear cracks^{7}. Frictional sliding is initiated only when a rupture traverses the entire interface^{1}; however, arrested ruptures can occur at applied shears far below the transition to frictional motion^{8,9,10,11,12,13,14,15,16,17}. Here we show, by measuring the real contact area and strain fields near rough frictional interfaces, that fracture mechanics quantitatively describe rupture arrest and therefore determine the onset of overall frictional sliding. Our measurements reveal both the local dissipation and the global elastic energy released by the rupture. The balance of these quantities entirely determines rupture lengths, whether finite or systemwide. These results confirm a fracturemechanicsbased paradigm^{7,15,18} for describing frictional motion and shed light on the selection^{18,19,20,21} of an earthquake’s magnitude.
Main
A frictional interface is formed by the interlocked solid asperities of rough surfaces in contact, whose area is much smaller than the nominal one^{6}. Failure of the asperities via rupture fronts is the fundamental mechanism responsible for the transition from static to sliding frictional motion^{1,4}. Ruptures can propagate well before the onset of global sliding, and then arrest before spanning the entire interface. Rupture arrest can, for example, result from inhomogeneous stress distributions along the interface^{8,9,10}. As no overall motion of the contacting bodies is induced by such events, they are often called precursors to sliding motion. Arrested events are analogous to earthquakes, which are dynamic ruptures of finite extent within preexisting natural faults; the boundary between contacting tectonic plates^{22,23}. Predicting the length of precursory ruptures is, therefore, closely related to the question of what determines the size of an earthquake^{18,19,20,21}.
Since their initial discovery^{8}, a rich variety of models has been dedicated to the dynamics of precursory ruptures in frictional systems. Aimed at reproducing nucleation and arrest, these include minimalistic onedimensional (1D) models^{9}, discrete contacts descriptions^{11,13,16}, rateandstate friction laws^{17} and fracture mechanics^{12}. These models are able to reproduce the existence of arrested ruptures but they provide no explicit predictions of where and how arrest occurs in real systems. Recent theoretical work^{15} explained the available data^{8} by explicitly demonstrating how fracture mechanics can be used to predict rupture arrest. Here we describe new experiments that confirm these theoretical predictions and show that this general framework indeed enables us to understand the selection of rupture length for any system geometries and loading conditions.
Recent experiments have shown that the strain fields driving ruptures along frictional interfaces are described^{7} by the linear elastic fracture mechanics (LEFM; ref. 24). In this context, a precursory event is an arrested crack^{12,15,18}. In fracture mechanics, crack arrest is defined by the Griffith criterion^{24}, a crack arrests when the amount of energy flowing to its tip becomes smaller than the fracture energy Γ, the dissipated energy per unit crack advance. Following the theoretical approach outlined in earlier studies^{15,18} we will experimentally verify that this criterion is fulfilled by spontaneously arrested frictional ruptures. These results provide new insights into the predictability of both frictional processes and earthquake dynamics.
We examine the frictional sliding of two poly(methylmethacrylate) (PMMA) blocks whose contact interface is flat to within a few μm. We define x, y and z as, respectively, the rupture propagation, normal loading and sample thickness directions. We use two different sample geometries (Fig. 1a). The asymmetric geometry is formed by blocks of different thicknesses (6 mm top block, 30 mm bottom block) and dimensions with a 3 μm r.m.s. surface roughness along the interface. The symmetric geometry is made of blocks having the same dimensions whose interface is formed by two optically flat surfaces. PMMA has a strainratedependent Young’s modulus 3 < E < 5.6 GPa and Poisson ratio v = 1/3. The Rayleigh wave speeds are c_{R} = 1,237 ± 10 m s^{−1} (plane stress) and c_{R} = 1,255 ± 10 m s^{−1} (plane strain) (Methods). The blocks are pressed together with an externally imposed normal force F_{N}, ranging from 1,500–7,000 N. After applying F_{N}, shear forces F_{S} are applied either at a single point or more uniformly via a sliding stage (Fig. 1a), giving rise to a variety of inhomogeneous distributions of normal and shear stresses along the interface (see examples in Fig. 1b). Throughout each experiment, continuous parallel measurements of the 2D strain tensor ɛ_{ij}(t) are performed at 15 to 18 locations along and 3.5 mm above the interface at a rate of 10^{6} samples per second (Fig. 1a). From the strain measurements we obtain stresses, while taking into account material viscoelasticity (Methods). At the same time, we measured the real contact area A(x, t) at 1,280 × 8 (x × y) spatial locations along the interface at 580,000 frames per second by an optical method based on total internal reflection^{1,4,7}. Under these stress conditions, we observe a succession of arrested ruptures of increasing length (Fig. 1c), well before F_{S} reaches the threshold for overall stick–slip motion of the blocks (Fig. 1d). The transition to the overall motion of the contacting blocks happens only when these ruptures span the entire interface (event VI).
The variations of the stresses at the tip of each propagating rupture are quantitatively described by the singular fields predicted by LEFM for shear (mode II) cracks^{7} (Fig. 2a). The singular term of the stress field has the form^{24}: in polar coordinates with respect to the crack tip, where Σ_{ij}^{II}(θ, v) is a known universal angular function and K_{II} is the stress intensity factor. Δσ_{ij} expresses the stress changes^{25} between the initially applied and residual stresses along the frictional crack faces. In the framework of LEFM no generality is lost by adding constant values, σ_{xx}^{0}, σ_{yy}^{0} and σ_{xy}^{res} (Fig. 2a) to equation (1) (Methods). For a known rupture velocity v, the amplitude of the singular term—that is, the stress intensity factor K_{II}—is related to the energy release rate G, defined as the flux of elastic energy per unit extension of a crack’s tip, by: where f_{II}(v) is a universal function, whose value is nearly unity for low rupture velocities v. The coefficient is α = 1 for the plane stress conditions used in the symmetric system, whereas α = (1 − v^{2}) for the plane strain conditions used in the asymmetric system^{7}. Following equation (1), fitting the three stress components (see, for example, Fig. 2a) provides a dynamic measurement^{7} of K_{II}.
During rupture propagation, G = Γ. Using the value of K_{II} in equation (2), we can therefore calculate the fracture energy Γ, which has been shown to be roughly independent of velocity^{7}. As Fig. 2b shows, we find that Γ depends linearly on the mean normal stress, 〈σ_{yy}〉. If we assume that the real contact area, A, entirely determines Γ, this result is in accordance with both the Bowden and Tabor picture^{6} (where A ∝ F_{N}) and the suggestion^{7} that Γ effectively measures the number of contacts that have to be broken in order for a crack to propagate.
Can a frictional rupture arrest be described by the fracturemechanicsbased criterion for crack arrest? If so, according to equation (2), on rupture arrest, G → G_{stat}(l) = α(K_{II}^{2}(l, v = 0)/E) and arrest will occur^{24} when G_{stat}(l) ≤ Γ. In general, K_{II}(l, v = 0) ≡ K_{II}^{stat}(l), can be explicitly calculated^{24} providing G_{stat}(l). For our sample geometry^{15,26} (Methods): where F(s/l) = 1 + 0.3(1 − (s/l)^{5/4}) and the stress drop for each event is defined as Δτ(x) = σ_{xy}^{0}(x) − σ_{xy}^{res}(x); the stress drop from the initial shear stress σ_{xy}^{0}(x) to the residual stress σ_{xy}^{res} at each point (Fig. 2a).
To predict the arrest location using equation (3), we need to measure both σ_{xy}^{0}(x) and σ_{xy}^{res}(x). Figure 2c, d demonstrates that both quantities vary significantly in space. By definition, σ_{xy}^{res}(x) is not defined beyond the rupture arrest location. Except very near x = 0, where applied torques produce measurable effects^{27}, we find that, within a given experiment, σ_{xy}^{res}(x) is invariant from one event to another (Fig. 2d). After accounting for the edge effects (Methods), we therefore use σ_{xy}^{res}(x) from the first systemwide event, as characteristic of the interface. We can now measure Δτ(x) for each event at each point x (Fig. 3a). As stresses are measured slightly above the interface, we improve the accuracy of Δτ(x) at the interface by accounting for stress gradients (Methods).
In Fig. 3a we also present the computed value of K_{II}^{stat}(l) (see equation (3)) in successive events for the experiment described in Fig. 1. Predicted arrest locations, ℓ_{predicted}, for each event are the locations where G_{stat} = Γ, where Γ is determined by 〈σ_{yy}〉, the averaged value of σ_{yy} in the section of the interface where ruptures arrest. In Fig. 3b we compare ℓ_{predicted} with the measured arrest length, ℓ_{measured}, obtained from the contact area measurements. ℓ_{predicted} agrees well with ℓ_{measured}. The same procedure is applied for 11 other experiments, each includes from 4 to 9 precursory events. We performed each experiment under different loading conditions for a wide range of Γ (Fig. 4). It is clear that all of the predicted arrest lengths are in excellent agreement with the measured lengths. The variety of stress distributions and values of Γ that are used emphasize the generality of this result. Note that this framework naturally incorporates the effects of local stress variations. The dependence of the arrest location with a heterogeneous distribution of stresses is not trivial; a given mean value of Δτ(x) can yield large variations in the predicted rupture arrest location, as the expression that determines G_{stat}(l) generally includes a singular weight function^{24} (see, for example, equation (3)). For example, stress fluctuations more strongly affect the arrest if located near the arrest point.
Our results provide clear evidence that frictional rupture is really a fracture process that can be quantitatively described by fracture mechanics. The concepts presented here suggest a completely different paradigm for understanding friction from that of the classical picture, which is based on the balance of local forces (stresses). We have shown that rupture arrest is governed by energy balance precisely as described by the Griffith criterion; balancing the global (systemwide) release of elastic energy, which is embodied in the integral nature of equation (3), and the local dissipative properties of the interface governed by Γ.
Although in our experiments we considered relatively constant values of Γ, energy balance is also valid when Γ varies along the interface. Locally high values of Γ, can, for instance, act as a barrier to propagation. What can cause spatial variations of Γ? We have shown that Γ ∝ σ_{yy} as the normal stress governs the geometrical size of the real contact area. Furthermore, Γ could be affected by varying interface properties—for example, interfaces incorporating different materials, chemical treatments, or pressureinduced phase transitions.
We have shown that applying fracture mechanics to friction has fundamental predictive power; knowledge of the prospective stress release at each point along the interface will tell us where a frictional rupture will arrest and, moreover, when ruptures will traverse the entire interface and precipitate frictional sliding. The initial stress profile along the interface, coupled with knowledge of the residual stress, entirely determines the eventual rupture length. Two caveats currently impede making specific predictions: knowledge of the residual stresses along the interface before an upcoming event and knowledge of when rupture nucleation will occur. The first of these might well be addressed by knowledge of the normal stress profile along the interface. Residual stresses are the manifestation of the nonbroken contacts that sustain the normal load at the frictional interface. In preliminary work, we observe a strong correlation between residual shear stress and local normal stress, suggesting a local dynamic friction coefficient, but additional study is required to cement this relationship. Rupture nucleation, or the onset of friction, is a more delicate point. Previous work has shown^{5,27} that characteristic static friction coefficients that govern the onset of frictional motion do not exist. The Griffith criterion could, as in rupture arrest, predict rupture nucleation. The Griffith criterion, however, can be applied only when a welldefined crack tip exists. In the case of rupture arrest, a singular tip obviously exists before arrest^{7}. Before nucleation, however, no initially sharp crack exists; a sharp initial crack must either be created or be dynamically formed. This enigmatic nucleation stage is still the subject of much active research^{20,23,28,29,30}.
Although the results described above are generally relevant to every frictional interface, they are especially important to the particular question of earthquake arrest. Understanding what determines an earthquake’s spatial extent is a central unresolved issue^{18,19,21}; either the rupture size is selected during the nucleation process or an earthquake arrests only when encountering a sufficiently high barrier. We have shown that the selection of the rupture length is deterministic; balancing the global stress release with local dissipation. Our results enable us to understand both viewpoints. When Γ does not vary significantly, the initial stress profile will wholly govern rupture arrest. For example, a large drop of stress near the nucleation zone can control the arrest location. On the other hand, the existence of a sufficiently high local value of Γ could precipitate rupture arrest by overcoming the dominance of the stress profile.
To conclude, it is far from trivial that fracture mechanics necessarily govern rupture arrest in experiments. A text book shear crack is in many ways different from frictional ruptures in a real experimental system which include significant normal stress variations, nonconstant initial and residual shear stresses and effects of frictional dissipation along the crack faces. Despite these numerous and substantial complexities, fracture mechanics still provide a fundamental description of friction which is surprisingly accurate.
Methods
Experimental setup.
We used blocks of poly(methylmethacrylate) (PMMA) with a Young’s modulus E_{s} = 3 GPa for low strain rates and E_{d} = 5.6 GPa for high strain rates (see the next section) and a Poisson ratio v = 1/3. Measured material wave speeds are: shear waves c_{S} = 1,345 ± 10 m s^{−1}, longitudinal waves c_{L} = 2,700 ± 10 m s^{−1}. These provide Rayleigh wave speeds c_{R} = 1,237 ± 10 m s^{−1} for plane stress conditions and c_{R} = 1,255 ± 10 m s^{−1} for plane strain conditions.
Symmetric setup. Blocks dimensions are 200 mm × 100 mm × 5.5 mm (x, y, z directions). The contacting surfaces are optically flat. The upper surface of the top block was fixed, while the bottom block was sheared using a pushrod of dimensions 3.5 mm × 5.5 mm (y × z directions), positioned at −3.5 mm < y < 0 and x = 0 mm. In addition, an optional rigid stopper (of crosssection 12 mm × 12 mm) was pressed against the top block, at x = 200 mm and y = 11 mm, to constrain the motion of this edge and, thereby, impose stress gradients in the x and y directions while a pushrod applied shear to the opposite edge. Strain field measurements were performed at 18 locations along the interface for 5.5 mm < x < 158.1 mm, and y = 3.5 mm above the interface. Rosettetype strain gauges were spaced 7.5 mm apart on average (6.5 mm < d < 8.5 mm).
Asymmetric setup. The asymmetric setup was constructed with a bottom block of dimensions 290 mm × 28 mm × 30 mm and a top block of dimensions 150 mm × 100 mm × 6 mm (x, y, z directions). The surface of the bottom block had a 3 μm r.m.s. roughness whereas the surface of the top block surface was optically flat. The upper surface of the top block was fixed and the bottom block was sheared from below by a sliding stage. A stopper, with the same characteristics as described above, prevented the displacement of the top block. Strain field measurements were performed at 15 locations along the interface for 4.3 mm < x < 144 mm, and y = 3.5 mm above the interface. Rosette type strain gauges were spaced 10 mm apart on average (7.5 mm < d < 15 mm). The resolution of the strain gauges was approximately 20–60 μstrain, leading to a 0.1–0.3 MPa resolution in the computed stress.
Transformation from strain to stress.
Boundary conditions are chosen as plane stress (σ_{zz} = 0; that is, free material expansion in the z direction) for the symmetric setup and plain strain (ɛ_{zz} = 0; that is, no material expansion in the z direction) for the asymmetric setup^{7}. The poly(methylmethacrylate) (PMMA) is viscoelastic^{31}. We take this effect into account while transforming strains into stresses: the static loading stresses are calculated in terms of the static Young’s modulus E_{s} whereas the rapid drop of stress due to the rupture propagation is calculated using the dynamic Young’s modulus E_{d}. For plane stress conditions, we calculated stresses from strains as^{32}:
Calculation of the static stress intensity factor.
Equation (3) is adapted from equation (8.3) from Tada^{26}. Tada’s solution provides the static stress intensity factor for a preexisting crack of length l, in a plate with a free vertical boundary at x = 0, when stresses are applied to the crack’s faces. We obtain equation (3) by superposition^{24,26}; subtracting the measured prestress along the prospective crack’s path from Tada’s solution. As a result, equation (3) calculates the stress intensity factor for a virtual crack of length l, as a function of the preexisting stresses along the virtual crack’s path. This formulation is conceptually equivalent to the wellknown Eshelby’s integral (Freund’s equation (6.4.31); ref. 24) but incorporates the free boundary at x = 0.
Definition of residual stresses.
We have shown that the residual shear stress following a rupture event is roughly constant for a given experiment (Fig. 2d). Nevertheless, the top block rotates slightly during the shear loading^{27} and thereby induces a variation of the normal and residual stresses. This effect is largest at the interface corner x = 0. To account for this effect, we define the residual stress of a precursor event n as follows: where σ_{xy, n}^{res} and σ_{xy, N}^{res} are respectively the residual stress of the event n and the main event N (systemwide event); x_{1stSG} is the location of the first strain gauge along the interface (x_{1stSG} ∼ 5 mm for both setups). The stress is linearly extrapolated from x = 0 to x = x_{1stSG}. Using this definition, we established the stress that would be released by a crack for any x along the interface for each nth event, Δτ(x) = σ_{xy, n}^{pre}(x) − σ_{xy, n}^{res}(x) (Fig. 3a).
Taylor’s expansion of stresses on the interface.
As stresses are measured at y = 3.5 mm above the interface, and as strong gradients of stresses due to the inhomogeneous loading may take place, stresses above the interface are not a perfect measure of the stresses on the interface. To correct for the finite height of the strain gauge placement, we used a Taylor’s expansion to improve our onfault stress measurements. At equilibrium^{32}, ∂σ_{ik}/∂x_{k} = 0 ⇒ ∂σ_{xy}/∂y = −(∂σ_{xx}/∂x). This leads to σ_{xy}(0) = σ_{xy}(y_{SG}) + y_{SG}(∂σ_{xx}/∂x), where y_{SG} = 3.5 mm is the position of the strain gauges and y = 0 defines the interface. The correction is typically around 5–10% of the stress nominal value. The use of this correction does not significantly change the results, but it does reduce their scatter.
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Acknowledgements
We acknowledge support from the James S. McDonnell Fund, the European Research Council (Grant No. 267256) and the Israel Science Foundation (Grant 76/11). E.B. acknowledges support from the Lady Davis Trust. We thank G. Cohen for comments.
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The Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
 Elsa Bayart
 , Ilya Svetlizky
 & Jay Fineberg
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E.B. and I.S. performed the measurements. All authors contributed to the analysis and writing the manuscript.
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The authors declare no competing financial interests.
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Correspondence to Jay Fineberg.
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